**Derivatives of Absolute Value Functions Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on finding derivatives of absolute value functions.

Before look at the worksheet, if you would like to learn, how to find derivative of an absolute value function,

**Problem 1 : **

Differentiate |2x+1| with respect to x.

**Problem 2 :**

Differentiate |x^{3}+1| with respect to x.

**Problem 3 :**

Differentiate |x|^{3} with respect to x.

**Problem 4 :**

Differentiate |2x-5| with respect to x.

**Problem 5 :**

Differentiate (x-2)^{2} + |x-2| with respect to x.

**Problem 6 :**

Differentiate 3|5x+7| with respect to x.

**Problem 7 :**

Differentiate |sinx| with respect to x.

**Problem 8 :**

Differentiate |cosx| with respect to x.

**Problem 9 :**

Differentiate |tanx| with respect to x.

**Problem 10 :**

Differentiate |sinx + cosx| with respect to x.

**Problem 1 : **

Differentiate |2x+1| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|2x+1|' = [(2x+1)/|2x+1|] ⋅ (2x+1)'

|2x+1|' = [(2x+1)/|2x+1|] ⋅ 2

|2x+1|' = 2(2x+1) / |2x+1|

**Problem 2 :**

Differentiate |x^{3}+1| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|x^{3}+1|' = [(x^{3}+1)/|x^{3}+1|] ⋅ (x^{3}+1)'

|x^{3}+1|' = [(x^{3}+1)/|x^{3}+1|] ⋅ 3x^{2}

|x^{3}+1|' = 3x^{2}(x^{3}+1) / |x^{3}+1|

**Problem 3 :**

Differentiate |x|^{3} with respect to x.

**Solution :**

In the given function |x|^{3}, using chain rule, first we have to find derivative for the exponent 3 and then for |x|.

(|x|^{3})' = {3|x|^{2}} ⋅ [x/|x|] ⋅ (x)'

(|x|^{3})' = {3|x|^{2}} ⋅ [x/|x|] ⋅ (1)

(|x|^{3})' = 3x|x|

**Problem 4 :**

Differentiate |2x-5| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|2x-5|' = [(2x-5)/|2x-5|] ⋅ (2x-5)'

|2x-5|' = [(2x-5)/|2x-5|] ⋅ 2

|2x-5|' = 2(2x-5) / |2x-5|

**Problem 5 :**

Differentiate (x-2)^{2} + |x-2| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

{(x-2)^{2} + |x-2|}' = [(x-2)^{2}]' + |x-2|'

{(x-2)^{2} + |x-2|}' = 2(x-2) + [(x-2)/|x-2|] ⋅ (x-2)'

{(x-2)^{2} + |x-2|}' = 2(x-2) + [(x-2)/|x-2|] ⋅ (1)

{(x-2)^{2} + |x-2|}' = 2(x-2) + (x-2) / |x-2|

**Problem 6 :**

Differentiate 3|5x+7| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

3|5x+7|' = 3 ⋅ [(5x+7)/|5x+7|] ⋅ (5x+7)'

3|5x+7|' = 3 ⋅ [(5x+7)/|5x+7|] ⋅ 5

3|5x+7|' = 15(5x+1) / |5x+7|

**Problem 7 :**

Differentiate |sinx| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|sinx|' = [sinx/|sinx|] ⋅ (sinx)'

|sinx|' = [sinx/|sinx|] ⋅ cosx

|sinx|' = (sinx ⋅ cosx) / |sinx|

**Problem 8 :**

Differentiate |cosx| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|cosx|' = [cosx/|cosx|] ⋅ (cosx)'

|cosx|' = [cosx/|cosx|] ⋅ (-sinx)

|cosx|' = - (sinx ⋅ cosx) / |cosx|

**Problem 9 :**

Differentiate |tanx| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|tanx|' = [tanx/|tanx|] ⋅ (tanx)'

|tanx|' = [tanx/|tanx|] ⋅ sec^{2}x

|tanx|' = sec²x ⋅ tanx / |tanx|

**Problem 10 :**

Differentiate |sinx + cosx| with respect to x.

**Solution :**

Using the formula of derivative of absolute value function, we have

|sinx + cosx|' = [(sinx+cosx) / |sinx+cosx|] ⋅ (sinx+cosx)'

|sinx + cosx|' = [(cosx+sinx) / |sinx+cosx|] ⋅ (cosx-sinx)

|sinx + cosx|' = (cos^{2}x - sin^{2}x) / |sinx+cosx|

|sinx + cosx|' = cos2x / |sinx+cosx|

After having gone through the stuff given above, we hope that the students would have understood, how to find derivative of an absolute value function.

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